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Simultaneous Blind Demixing and Super-resolution via Vectorized Hankel Lift

Haifeng Wang, Jinchi Chen, Hulei Fan, Yuxiang Zhao, Li Yu

TL;DR

This work tackles simultaneous blind demixing and super-resolution under a subspace model by reformulating the problem as a structured low-rank matrix demixing task. It introduces MVHL, a convex program based on vectorized Hankel lifting and nuclear-norm minimization, to jointly demix and super-resolve multiple signal components from linear measurements. The authors prove exact recovery under standard incoherence assumptions when $n \gtrsim K \mu_0 \mu_1 s r \log(sn)$, using a dual certificate constructed via a golfing scheme. Empirical results show MVHL outperforms ANM in phase transitions, exhibits robustness to noise, and enables accurate joint radar-communications parameter estimation when combined with MUSIC.

Abstract

In this work, we investigate the problem of simultaneous blind demixing and super-resolution. Leveraging the subspace assumption regarding unknown point spread functions, this problem can be reformulated as a low-rank matrix demixing problem. We propose a convex recovery approach that utilizes the low-rank structure of each vectorized Hankel matrix associated with the target matrix. Our analysis reveals that for achieving exact recovery, the number of samples needs to satisfy the condition $n\gtrsim Ksr \log (sn)$. Empirical evaluations demonstrate the recovery capabilities and the computational efficiency of the convex method.

Simultaneous Blind Demixing and Super-resolution via Vectorized Hankel Lift

TL;DR

This work tackles simultaneous blind demixing and super-resolution under a subspace model by reformulating the problem as a structured low-rank matrix demixing task. It introduces MVHL, a convex program based on vectorized Hankel lifting and nuclear-norm minimization, to jointly demix and super-resolve multiple signal components from linear measurements. The authors prove exact recovery under standard incoherence assumptions when , using a dual certificate constructed via a golfing scheme. Empirical results show MVHL outperforms ANM in phase transitions, exhibits robustness to noise, and enables accurate joint radar-communications parameter estimation when combined with MUSIC.

Abstract

In this work, we investigate the problem of simultaneous blind demixing and super-resolution. Leveraging the subspace assumption regarding unknown point spread functions, this problem can be reformulated as a low-rank matrix demixing problem. We propose a convex recovery approach that utilizes the low-rank structure of each vectorized Hankel matrix associated with the target matrix. Our analysis reveals that for achieving exact recovery, the number of samples needs to satisfy the condition . Empirical evaluations demonstrate the recovery capabilities and the computational efficiency of the convex method.
Paper Structure (16 sections, 6 theorems, 47 equations, 3 figures)

This paper contains 16 sections, 6 theorems, 47 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Problem formulation and proposed method
  3. Problem Formulation
  4. Proposed Method
  5. Main results
  6. Simulation Results
  7. Recovery Ability of MVHL
  8. Robustness of MVHL
  9. Channel Parameter Estimation
  10. Proofs of Main Result
  11. Deterministic Optimality Condition
  12. Construction of Dual Certificate
  13. Validating the Dual Certificate and Completing the Proof
  14. Validating \ref{['condition F norm']}
  15. Validating \ref{['condition O norm']}
  16. ...and 1 more sections

Key Result

Theorem 3.1

Under Assumptions assumption 1 and assumption 2, the matrices $\{ \bm{X}_k^\natural \}_{k = 1}^K$ are the unique optimal solution to the problem eq: oripro with probability at least $1-c_0(sn)^{-c_1}$, provided that $n\gtrsim K\mu_0\mu_1sr\log(sn)$, where $c_0,c_1$ are absolute constants.

Figures (3)

  • Figure 1: The phase transitions of MVHL and ANM
  • Figure 2: Performance of MVHL under different noise levels
  • Figure 3: Performance of MVHL for channel parameter estimation problem

Theorems & Definitions (9)

  • Remark 3.1
  • Remark 3.2
  • Theorem 3.1
  • Theorem 5.1
  • Lemma 5.2
  • proof : Proof of Theorem \ref{['thm: unique theorem']}
  • Lemma 5.3
  • Lemma 5.4
  • Corollary 5.5